Finite Group Databasebeta

← #10708907

Subgroups — #10708907

Order 768.

288 conjugacy classes of subgroups; only the 73 normal subgroups are shown.

normal subgroups only

This group has 288 conjugacy classes of subgroups — too many to draw. Only its 73 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1Q8×Dic12C211yesmaximal, characteristic
2#70942C211yesmaximal, characteristic
3#69293C211yesmaximal, characteristic
4#69065C211yesmaximal, characteristic
5C3×Q8⋊C16C211yesmaximal, characteristic
6#67565C211yesmaximal, characteristic
7#67505C211yesmaximal, characteristic
8#12358—31nomaximal, Sylow
9Q8×Dic6—21no—
10C3×C8×Q8C2211yescharacteristic
11#5020—21no—
12C2×Dic24—21no—
13#4850—21no—
14#4850—22no—
15#4699C2211yescharacteristic
16C4×Dic12—21no—
17C4×Dic12C2211yescharacteristic
18#4682—21no—
19#4682C2211yescharacteristic
20C3×C4⋊C16C2211yescharacteristic
21C3×C4×C16C2211yescharacteristic
22#4503—21no—
23#4503—22no—
24#4482—21no—
25#4458C2211yescharacteristic
26Q8×Q16—31no—
27#2241—31no—
28#2209—31no—
29#1670—31no—
30#1365—31no—
31Q8⋊C16D311yescharacteristic
32#1332—31no—
33C3×C4×Q8D411yescharacteristic
34Q8×Dic3—41no—
35C2×Dic12—21no—
36C2×Dic12—21no—
37C2×Dic12D411yescharacteristic
38#888—42no—
39#888—41no—
40#869—22no—
41#869—21no—
42#869—21no—
43C4×Dic6—22no—
44C4×Dic6—21no—
45C2×C3×C16—21no—
46C2×C3×C16—21no—
47C3×C4⋊C8—21no—
48C3×C4⋊C8D411yescharacteristic
49C3×C4×C8—21no—
50C3×C4×C8C2311yescharacteristic
51#835—41no—
52#818—22no—
53#818—21no—
54#818D411yescharacteristic
55#818D412yes—
56#816—22no—
57#816—21no—
58#808—41no—
59C4⋊(C3⋊C8)—21no—
60Dic24—42no—
61Q82—61no—
62C2×Q32—61no—
63#500—31no—
64#500—61no—
65#494—31no—
66#477—62no—
67#477—61no—
68#475—61no—
69C8×Q8C2×D311yescharacteristic
70C4×Q16—31no—
71C4×Q16—61no—
72C4⋊C16C2×D311yescharacteristic
73#358—61no—
74#358—62no—
75C4×C16C2×D311yescharacteristic
76#335—31no—
77#332—61no—
78C2×C3×Q8D811yescharacteristic
79C2×Dic6—21no—
80C2×Dic6—41no—
81C2×Dic6—42no—
82C2×Dic6—41no—
83C2×C3×C8—21no—
84C2×C3×C8C2×D411yescharacteristic
85C2×C3×C8C2×D411yescharacteristic
86C3×C4⋊C4—21no—
87C3×C4⋊C4D811yescharacteristic
88C3×C42—21no—
89C3×C42C2×D411yescharacteristic
90C4⋊Dic3—22no—
91C4⋊Dic3—21no—
92C4⋊Dic3—42no—
93C4⋊Dic3—41no—
94C4⋊Dic3—41no—
95C4⋊Dic3—41no—
96Dic3⋊C4—42no—
97Dic3⋊C4—41no—
98C4×Dic3—42no—
99C4×Dic3—41no—
100C2×C3⋊C8—41no—
101Dic12—21no—
102Dic12—42no—
103Dic12—41no—
104Dic12—41no—
105Dic12SD1612yes—
106C3×C16—41no—
107C3×C16—41no—
108C2×Q16—31no—
109C2×Q16—61no—
110C2×Q16—61no—
111C4⋊Q8—62no—
112C4⋊Q8—61no—
113C4⋊Q8—61no—
114C4⋊Q8—122no—
115C4⋊Q8—121no—
116C4×Q8—62no—
117C4×Q8—61no—
118C4×Q8—121no—
119C4×Q8D1211yescharacteristic
120Q32—122no—
121C2×C16—21no—
122C2×C16—21no—
123#107—31no—
124#107—32no—
125#107—62no—
126#107—61no—
127#106—121no—
128C4⋊C8—21no—
129C4⋊C8—61no—
130C4⋊C8D1211yescharacteristic
131#103—62no—
132#103—61no—
133#103—121no—
134C4×C8—21no—
135C4×C8C22×D311yescharacteristic
136C3×Q8—21no—
137C3×Q8Q3212yescomplemented
138C2×C3×C4—21no—
139C2×C3×C4—21no—
140C2×C3×C4#12011yescharacteristic
141C2×C3×C4C2×D811yescharacteristic
142C2×C3×C4AGL(1,ℤ/8)11yescharacteristic
143C2×Dic3—41no—
144C2×Dic3—42no—
145C2×Dic3—41no—
146C2×Dic3—81no—
147Dic6—22no—
148Dic6—41no—
149Dic6—81no—
150Dic6—81no—
151Dic6—161no—
152Dic6—162no—
153C3×C8—21no—
154C3×C8—41no—
155C3×C8C2×SD1611yescharacteristic
156C3×C8#13711yescharacteristic
157C3⋊C8—81no—
158C2×Q8—61no—
159C2×Q8—122no—
160C2×Q8—121no—
161C2×Q8—121no—
162C2×Q8D2411yescharacteristic
163Q16—32no—
164Q16—61no—
165Q16—122no—
166Q16—121no—
167Q16—121no—
168C2×C8—21no—
169C2×C8—121no—
170C2×C8D3×D411yescharacteristic
171C2×C8D3×D411yescharacteristic
172C4⋊C4—21no—
173C4⋊C4—62no—
174C4⋊C4—61no—
175C4⋊C4—122no—
176C4⋊C4—122no—
177C4⋊C4—121no—
178C4⋊C4—121no—
179C4⋊C4—121no—
180C4⋊C4—121no—
181C4⋊C4D2411yescharacteristic
182C42—21no—
183C42—122no—
184C42—121no—
185C42C2×D1211yescharacteristic
186C16—41no—
187C16—41no—
188C22×C3#44711yescharacteristic
189C3×C4—22no—
190C3×C4—21no—
191C3×C4—41no—
192C3×C4#44811yescharacteristic
193C3×C4#45411yescharacteristic
194C3×C4C2×Q3211yescharacteristic
195C3×C4#51011yescharacteristic
196Dic3—42no—
197Dic3—82no—
198Dic3—81no—
199Dic3—161no—
200Q8—21no—
201Q8—62no—
202Q8—121no—
203Q8—241no—
204Q8—241no—
205Q8—481no—
206Q8—482no—
207Q8Dic2412yescomplemented
208C2×C4—21no—
209C2×C4—21no—
210C2×C4—121no—
211C2×C4—122no—
212C2×C4—121no—
213C2×C4—241no—
214C2×C4#88211yescharacteristic
215C2×C4C2×D2411yescharacteristic
216C2×C4#90811yescharacteristic
217C8—21no—
218C8—41no—
219C8—241no—
220C8D3×SD1611yescharacteristic
221C8#91811yescharacteristic
222C2×C3#162711yescharacteristic
223C2×C3#218811yescharacteristic
224C2×C3#219411yescharacteristic
225C22#473211yescharacteristic
226C4—21no—
227C4—22no—
228C4—41no—
229C4—122no—
230C4—242no—
231C4—241no—
232C4—481no—
233C4#480611yescharacteristic
234C4#482711yescharacteristic
235C4C2×Dic2411yescharacteristic
236C4#490911yescharacteristic
237C3#1235811yescharacteristic, complemented, Sylow
238C2#6826611yescharacteristic
239C2#6911311yescharacteristic
240C2#6912811yescharacteristic
subgroupG/H
centerC22#4732
commutator, cosocleC3×C4×C8C23
FrattiniC4×C8C22×D3
FittingC3×Q8⋊C16C2
socle, prime radicalC22×C3#447
hypercenterQ8⋊C16D3
hypocenterC3#12358
LCSG ▷ C3×C4×C8 ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C42 ◁ C8×Q8 ◁ Q8⋊C16

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
137. C3×Q8Q32semidirect
207. Q8Dic24semidirect
237. C3#12358semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
Q8×Dic122
#709422
#692932
#690652
C3×Q8⋊C162
#675652
#675052
#123583